Stochasticity is inherent in many systems. It might arise as the result of actuator effects, sensor readings, rate of arrivals, component failure rates, etc. Even though testing is a commonly used approach to verify systems, it relies on the ability of the engineers to write out test cases that cover all the behaviors of the system where the expected failures can occur. That is usually a very difficult task. Furthermore, in many cases, system failures can occur in unexpected operating conditions and inputs.
The verification problem for stochastic or probabilistic hybrid systems has received much attention over the years. The focus of the early work was mainly on probabilistic reachability verification problems. Temporal logic model checking and verification problems for stochastic hybrid systems have also been researched for a long time, and this is still an active research area. However, many of these works are focused on building system abstractions that can be checked using existing verification tools.
One issue with falsification algorithms used in model verification schemes is determining the number of tests required to verify a model. The number of tests required to verify a model often depends on the robustness landscape over the search space of the hybrid system. The discontinuities induced by the hybrid system may make it hard to locate the regions of interest for testing purposes.
In the falsification problem, the search space consists of initial conditions, input signals and other system parameters. However, the algorithm performance, i.e., how many tests are required before a bad behavior is detected, depends on the landscape induced by the specification robustness over the search space of the hybrid system. In particular, these methods apply more effectively to continuous (non-hybrid) dynamical systems. Even though continuous systems may still have a large number of local minima, their basin of attraction is usually large. The latter fact helps algorithms to locate faster regions of interest in the search space.
However, the same does not hold for hybrid systems. Local minima can now appear anywhere in the search space with very small or non-smooth basins of attraction. Therefore, the probability that these regions will be detected becomes very small. In addition, gradient descent algorithms will not work in such cases. However, if a system can extract information about potential discontinuities in the hybrid system, then the system can narrow a search to such regions which otherwise would have been rather improbable to sample from.